Integral geometry of exceptional spheres
| Autor: | Gil Solanes, Thomas Wannerer |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Mathematics - Differential Geometry
Pure mathematics Algebra and Number Theory Group (mathematics) 010102 general mathematics Metric Geometry (math.MG) 53C65 Curvature 01 natural sciences Action (physics) Integral geometry Differential Geometry (math.DG) Mathematics - Metric Geometry Complex space Simple (abstract algebra) FOS: Mathematics Tangent space Geometry and Topology 0101 mathematics Invariant (mathematics) Analysis Mathematics |
| Zdroj: | J. Differential Geom. 117, no. 1 (2021), 137-191 |
| ISSN: | 0022-040X |
| DOI: | 10.4310/jdg/1609902019 |
| Popis: | The algebras of valuations on $S^6$ and $S^7$ invariant under the actions of $\mathrm G_2$ and $\mathrm{Spin}(7)$ are shown to be isomorphic to the algebra of translation-invariant valuations on the tangent space at a point invariant under the action of the isotropy group. This is in analogy with the cases of real and complex space forms, suggesting the possibility that the same phenomenon holds in all Riemannian isotropic spaces. Based on the description of the algebras the full array of kinematic formulas for invariant valuations and curvature measures in $S^6$ and $S^7$ is computed. A key technical point is an extension of the classical theorems of Klain and Schneider on simple valuations. |
| Databáze: | OpenAIRE |
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